1. Field of the Invention
The present invention relates generally to sampling frequency converters, and more particularly, to a sampling frequency converter receiving digital data sampled at a given sampling rate for converting the same to digital data sampled at a higher sampling rate and a method therefor.
2. Description of the Prior Art
Digital processing of signals has some advantages, such as smaller dependency on characteristics of devices and/or elements and facility of various processing of signals, as compared to analog processing. On the other hand, in the technical fields of video apparatus and units and audio apparatus and units, it is desired that analog video signals and analog audio signals are converted to digital data to perform various signal processing in order to improve a reproduced image and a reproduced voice and sound in quality. In order to meet such a requirement, digital video apparatus and units such as a digital television receiver and a digital video tape recorder and digital audio apparatus and units such as a compact disc player and a digital audio tape recorder have been devised and put into practice.
However, in these kinds of digital video apparatus and units and digital audio apparatus and units, different sampling frequencies are used in converting an analog signal to digital data in different apparatus and units or manufacturers. For example, in digital television receivers, a sampling frequency f.sub.s of color TV signals is selected to be 3f.sub.sc or 4f.sub.sc in many cases, where f.sub.sc denotes a frequency of a chrominance subcarrier of approximately 3.58 MHz. As the kind and the number of such digital apparatus and units are increased, the connection between the apparatus and units each having different sampling frequencies f.sub.s provides a problem to be solved. More specifically, as shown in FIG. 1, when a device A processes data having a sampling rate or frequency f.sub.n while a device B processes data having a sampling rate or frequency f.sub.m, it is necessary that a sampling rate (sampling frequency) of output (or input) data coincides with a sampling rate of a supplying unit (or receiving unit) in order to supply and receive data between the devices A and B. Thus, a sampling frequency converter C must be provided between the devices A and B. Description is now made on the principle of an operation and the structure of a conventional sampling frequency converter.
It is assumed that a one dimensional signal g(t) limited to a band W or less has a frequency spectrum shown in FIG. 2(a). According to the sampling theorem, if sample values . . ., g(-2T), g(-T), g(0), g(T), g(2T), . . . sampled at the interval of T=1/(2W), that is, sampled with a sampling frequency of f.sub.s =2W, are given, the original signal g(t) can be completely recovered from the following equation (1): ##EQU1## where, T=1/(2W), ##EQU2## A sampling function S(t) indicates an output generated when an impulse is applied to an ideal low pass filter having the band W.
FIG. 3 shows the sampling function S(t) and one example of a band limited waveform recovered in accordance with the equation (1).
Since a function (FIG. 2(c)) composed of only sample values is given by the product of the original signal g(t) (FIG. 2(a)) and a one dimensional Comb function .SIGMA..delta.(t-iT) (FIG. 2(b)), as shown in FIG. 2(c), the function can be expressed as: EQU g(t).multidot..SIGMA..delta.(t-iT) (2)
If Fourier transform is performed on the expression (2), then: ##EQU3## .circle. in the above equation (3) denotes convolution. The spectrum distribution of the equation (3) is shown in FIG. 2(d).
Consideration will be given to a case in which time-sequential digital data Q.sub.0, Q.sub.1, Q.sub.2, . . . obtained by sampling the original signal g(t) with a sampling frequency 3f.sub.sc are digitally processed to be converted to time-sequential data P.sub.0, P.sub.1, P.sub.2, . . . obtained by sampling the original signal g(t) with a sampling frequency 4f.sub.sc, as shown in FIG. 4. In this case, theoretically, setting the respective sample data Q.sub.0, Q.sub.1, Q.sub.2, . . . to g(0), g(T), g(2T), . . . (where T=1/(3f.sub.sc)), to be substituted in the equation (1), then g(t) is represented as follows: EQU g(t)=.SIGMA.Q.sub.i S(t-iT) (1)'
Then, if the times at which the sample data P.sub.0, P.sub.l, P.sub.2, are obtained, i.e., 0, T', 2T', . . . (where T'=1/(4f.sub.sc)) are substituted in the equation (1)', sample data P.sub.0, P.sub.1, P.sub.2, . . . having the sampling frequency 4f.sub.sc are obtained. However, the above described operation with an original sampling frequency and a sampling frequency after conversion being set to arbitrary values, respectively, is almost impossible and is not practical from a point of view of circuit configuration or the like because the operation includes an infinite number of times of product and addition.
On the other hand, when the original sampling frequency f.sub.m and the sampling frequency f.sub.n after conversion satisfy the integral ratio of m:n, the original data Q.sub.i and the data P.sub.j after conversion have a particular phase relation, so that the conversion can be simplified, which is described in, for example, an article by M. Achiba, entitled "An approach for digitally converting a sampling frequency for NTSC signals", IECE Japan Precedings of National Conference, March 1979, the lecture No. 1080.
More specifically, as shown in FIG. 4, considering a case in which the sampling frequency 3f.sub.sc is converted to the sampling frequency 4f.sub.sc, if the data Q.sub.i is interpolated by an interpolating filter which operates at the rate of the least common multiple 12f.sub.sc of the sampling frequencies 3f.sub.sc and 4f.sub.sc to convert the sampling frequency 3f.sub.sc to the sampling frequency 12f.sub.sc and then, the interpolated data Q.sub.i ' is further resampled with the sampling frequency 4f.sub.sc, then the data P.sub.j after conversion is obtained. When data obtained by sampling a sampling function of T=1/(3f.sub.sc) with the sampling frequency 12f.sub.sc are represented by Sh (h=0, .+-.1, .+-.2, . . .) as shown in FIG. 4(a), the following equation (4) is obtained as a general expression between the original data Q.sub.i and the data after conversion P.sub.j, based on the particular phase relation (P.sub.4k =Q.sub.3k): ##EQU4## where S.sub.h denotes an impulse response of the ideal low pass filter (which operates at the rate of frequency 12f.sub.sc) having the band W (=1/2T=3/2f.sub.sc), which is given by the following equation, as described above (see the equation (1)): ##EQU5## where S.sub.0 =1.
As clearly seen from the equation (4), conversion of the sampling frequency from 3f.sub.sc to 4f.sub.sc is performed by structuring the interpolating filter using four kinds of transversal type digital filers shown in the equation (4).
As shown in FIG. 5, the transversal type filter generally comprises a plurality of delay elements D connected in series, coefficient circuits C each for multiplying an output of each of the delay elements by a predetermined constant (tap coefficient) .alpha..sub.1 to .alpha..sub.n-1, and an adder S for summing up an output of each of the coefficient circuits.
Thus, assuming that the delay time Z.sup.-1 of each of the delay elements D is 1/(3f.sub.sc) and the 4 kinds of tap coefficients .alpha..sub.1 to .alpha..sub.n-1 of the coefficient circuits C are {S.sub.4l }, {S.sub.4l-3 },{S.sub.4l-6 } and {S.sub.4l-9 }, respectively, converted data {P.sub.4k }, {P.sub.4k+1 }, {P4k+2} and {P.sub.4k+3 } can be obtained from 4 kinds of filters, respectively. Thus, if such digital filters 1a to 1d are provided in parallel as shown in FIG. 6 and outputs of the digital filers la to 1d are sequentially switched at the rate of the frequency 4f.sub.sc, a converted data stream P.sub.j of the sampling frequency 4f.sub.sc is obtained. In FIG. 6, a terminal 3 receives the original data stream Q.sub.i. The digital filter 1a has the tap coefficient {S.sub.4l } and outputs the converted data P.sub.4k. The filter 1b has the tap coefficient {S.sub.4l-3 } and outputs the converted data P.sub.4k+1. The digital filter 1c has the tap coefficient {S.sub.4l-6 } and outputs the converted data P.sub.4k+2. The digital filter 1d has the tap coefficient {S.sub.4l-9 } and outputs the converted data P.sub.4k+3. A switch 2 sequentially switches the filters 1a to 1d at the rate of the frequency 4f.sub.sc, to supply an output from a selected filter to an output terminal 4.
If an infinite number of times of addition is performed to find the converted data P.sub.j as in the above equation (4), ideal conversion is achieved, so that exact converted data is obtained. However, performing an infinite number of times of addition is impossible on an actual circuit because it means that an infinite number of delay elements D and coefficient circuits C are provided in, for example, the structure shown in FIG. 5. The number of times of addition in the filers 1a to 1d are generally determined from frequency characteristics of the interpolating filter based on required accuracy for a filter, or the like.
On the other hand, if considered in a frequency region, the interpolating filters 1a to 1d are low pass filters for removing higher harmonic components caused by sampling of f.sub.s =3f.sub.sc and eliminating components folded into a base band caused by resampling at the rate of f.sub.s =4f.sub.sc, as shown in FIGS. 7A and 7B. In color TV signals, since the spectral luminous efficacy in the vicinity of a direct current and f.sub.sc region is high, gain in a frequency which is integral multiple of f.sub.sc must be strictly suppressed. The above described interpolating filters theoretically operate at the rate of the frequency 12f.sub.sc, so that it is required for the filters to considerably suppress characteristics in f=n'f.sub.sc (n'=0, 1, 2, . . . , 6). This is achieved by a frequency sampling filter.
A frequency characteristic H.sub.1 (f) of a transversal filter of linear phase having (2N+1) impulse responses h.sub.i is given by the following equation (7): ##EQU6## where h.sub.i =h.sub.-i Assuming that the sampling frequency is 12f.sub.sc in the equation (7), the following equation is obtained: ##EQU7##
Here, the exponential component is removed in the expression (7') for convenience. Thus, the filter, in which the above described frequency characteristic H.sub.1 (f) is 1 for f=0, f.sub.sc while being 0 for f=2f.sub.sc, 3f.sub.sc, . . ., 6f.sub.sc, can be structured from the above equation (7)' by a transversal filter having 13 taps in which N is a minimum of 6, i.e., a sampling interval of a frequency is f.sub.sc (represented by a broken line in FIG. 4C). On the other hand, when a 25th order transversal filter having 25 taps (impulse responses) in which the sampling interval is f.sub.sc /2 is required, the filter is found from a simultaneous equation of 25 elements such that N=12 in the equation (7)' (because h.sub.i =h.sub.-i is also a condition) (see FIG. 7C).
When an impulse response S.sub.k of the transversal filter having 25 taps is calculated by using the above described simultaneous equation of 25 elements, the following equation (8) is obtained as an equation corresponding to the equation (4): ##EQU8## The above equation (8) is obtained if the phase relation in the equation (4) is considered in impulse responses S.sub.-12 to S.sub.12. When the equation (8) is expressed by a determinant, the following equation (9) is obtained: ##EQU9## Thus, as obvious from the equation (9), the filters 1a to 1d can be structured by a seventh or sixth order transversal filter having as tap coefficients [S.sub.-12, S.sub.-8, . . ., S.sub.8, S.sub.12 ], [S.sub.-11, S.sub.-7, . . ., S.sub.5, S.sub.9 ], [S.sub.-10, S.sub.-6, . . ., S.sub.6, S.sub.10 ] and [S.sub.-9, S.sub.-5, . . ., S.sub.7, S.sub.11 ], respectively.
FIG. 8 shows one example of an impulse response (tap coefficient) of the 25th order transversal filter for converting the sampling frequency from 3f.sub.sc to 4f.sub.sc and respective tap coefficients of the digital filters 1a to 1d.
As described above, in the conventional sampling frequency converter, when the original sampling frequency f.sub.m and the sampling frequency f.sub.n after conversion are in the integral ratio of m:n, conversion from the sample data Q.sub.i of the sampling frequency f.sub.m to the sample data P.sub.j of the sampling frequency f.sub.n is performed based on the impulse response data S.sub.h obtained by sampling the sampling function of T=1/f.sub.m with a frequency which is the least common multiple of the frequencies f.sub.m and f.sub.n. In the actual circuit configuration, under consideration of frequency response characteristics of the filter, or the like, a sampling frequency converter is structured by using a parallel body of n transversal type digital filters and switching means sequentially and selectively passing outputs of the filters, which operate at the rate of the frequency f.sub.n, as expressed by the equation (9).
However, in the conventional structure, n digital filters are required. Thus, the number of parts constituting a sampling frequency converter is increased, so that the circuit configuration becomes complicated and expensive.